The proposed expression for y(Horse | x) places a Taylor expansion beside the hierarchy φ(x), ψ(φ), and z(ψ). Its terms describe β together with changes along Δβ, including a first-derivative term and a second-order term. This frames movement from a base class toward another class as a local manifold approximation, not as demonstrated recognition.
What the expansion contributes
In the supplied formulation, Δβ marks a proposed direction of change, while the derivatives describe how β changes with respect to z along that transition. The first-order term captures a local direction of change; the second-order term adds curvature information to the approximation. This is a way to describe a nearby transition in the representation, provided the quantities and their relationships are made precise.
Local means local
A Taylor expansion is a local description: its terms approximate behavior around a chosen point or region. Extending that description toward an unseen class therefore relies on the transition remaining well approximated along the path. The expression suggests that assumption, but does not establish how large a step is safe or whether the approximation holds for a particular representation.
Separate promise from performance
The hierarchy and expansion connect naturally to zero-shot learning because they offer a structured account of parts, semantic direction, and local change. Yet the material supplied here does not include prediction results, so no conclusion about practical performance follows. The useful takeaway is narrower: treat the expansion as a theoretical model of local adaptation, then ask separately whether its predictions hold for unseen classes.